Locally analytic vectors and decompletion in mixed characteristic

In p -adic Hodge theory and the p -adic Langlands program, Banach spaces with Qₚ -coefficients and p -adic Lie group actions are central. Studying the subrepresentation of G -locally analytic vectors, Wˡᵃ , is useful because Wˡᵃ can be studied via the Lie algebra Lie(G) , which simplifies the action of G . Additionally, Wˡᵃ often behaves as a decompletion of W , making it closer to an algebraic or geometric object. This article introduces a notion of locally analytic vectors for W in a mixed characteristic setting, specifically for Zₚ -Tate algebras. This generalization encompasses the classical definition and also specializes to super-Hölder vectors in characteristic p . Using binomial expansions instead of Taylor series, this new definition bridges locally analytic vectors in characteristic $0$ and characteristic p . Our main theorem shows that under certain conditions, the map W ↦ Wˡᵃ acts as a descent, and the derived locally analytic vectors Rₗₐⁱ(W) vanish for i ≥ 1 . This result extends Theorem C of [Por24], providing new tools for propagating information about locally analytic vectors from characteristic $0$ to characteristic p . We provide three applications: a new proof of Berger-Rozensztajn’s main result using characteristic $0$ methods, the introduction of an integral multivariable ring ALT,la in the Lubin-Tate setting, and a novel interpretation of the classical Cohen ring AQₚ from the theory of (φ ,Γ ) -modules in terms of locally analytic vectors.